A 50.5-g golf ball is driven from the tee with an initial speed of 57.4 m/s and rises to a height of 23.2 m. (a) Neglect air resistance and determine the kinetic energy of the ball at its highest point. (b) What is its speed when it is 7.97 m below its highest point?
Y^2 = Yo^2 + 2g*h = 0
Yo^2 = -2g*h = -2*(-9.8)*23.2 = 454.72
Yo = 21.32 m/s = Ver. component of initial velocity.
Tan A = Yo/Vo = 21.32/57.4 = 0.37150
A = 20.38o
a. Vo = 57.4m/s[20.4o]
Xo = 57.4*Cos20.4 = 53.8 m/s = Hor.
component of initial velocity.
V = Xo + Yi = 53.8 + 0 = 53.8 m/s at highest point.
KE = 0.5*M*V^2 = 0.5*0.0505*53.8^2 =
73.1 J.
b. Y^2 = Yo^2 + 2g*h=0 + 19.6(23.2-7.97)
= 298.5
Y = 17.3 m/s
V = Xo+Yi = 53.8 + 17.3i
V = sqrt(53.8^2+17.3^2)
Correction: Replace Tan A = Yo/Vo with
sin A = Yo/Vo = 21.32/57.4 = 0.37143
A = 21.8o = Launch angle.
a. Vo = 57.4m/s[21.8o]
Xo = 57.4*Cos21.8 = 53.3 m/s = Hor.
component of initial velocity.
V = Xo+Yi = 53.3 + 0 = 53.3 m/s at highest point.
KE = 0.5*M*V^2 = 0.5*0.0505*53.3^2 =
71.7 J.
b. Y^2 = Yo^2 + 2g*h=0 + 19.6(23.2-7.97)= 298.5
Y = 17.3 m/s
V = Xo+Yi = 53.3 + 17.3i
V = sqrt(53.3^2+17.3^2) =
To solve this problem, we can use the principle of conservation of energy. The total mechanical energy of the ball at any point in its trajectory is equal to the sum of its kinetic energy (KE) and potential energy (PE).
(a) To determine the kinetic energy of the ball at its highest point, we need to calculate its potential energy at the highest point and subtract it from its total mechanical energy.
The potential energy of an object at a certain height is given by the formula:
PE = m * g * h
Where:
m = mass of the golf ball = 50.5 g = 0.0505 kg (converting grams to kilograms)
g = acceleration due to gravity = 9.8 m/s^2
h = height = 23.2 m
Substituting the values into the formula:
PE = 0.0505 kg * 9.8 m/s^2 * 23.2 m
PE = 11.19 J (to two decimal places)
Since there is no air resistance, the total mechanical energy of the ball is conserved throughout its trajectory. Therefore, at its highest point, the kinetic energy (KE) is equal to:
KE_highest point = Total Mechanical Energy - Potential Energy
KE_highest point = Total Mechanical Energy - PE
KE_highest point = Total Mechanical Energy - 11.19 J
Now, let's calculate the initial total mechanical energy of the ball using its initial speed:
Total Mechanical Energy = KE_initial + PE_initial
The initial kinetic energy (KE_initial) of the ball is given by the formula:
KE_initial = (1/2) * m * v_initial^2
Where:
m = mass of the golf ball = 0.0505 kg
v_initial = initial speed = 57.4 m/s
Substituting the values into the formula:
KE_initial = (0.5) * 0.0505 kg * (57.4 m/s)^2
KE_initial ≈ 82.54 J (to two decimal places)
Now, we can calculate the total mechanical energy:
Total Mechanical Energy = KE_initial + PE_initial
Total Mechanical Energy ≈ 82.54 J + PE_initial
Since the potential energy at the highest point is the greatest, we can say that
Total Mechanical Energy = KE_highest point + 11.19 J
Let's solve for KE_highest point:
KE_highest point = Total Mechanical Energy - 11.19 J
KE_highest point ≈ 82.54 J - 11.19 J
Therefore, the kinetic energy of the ball at its highest point is approximately 71.35 J (to two decimal places).
(b) Now we need to determine the speed of the ball when it is 7.97 m below its highest point.
To solve for this, we will use the principle of conservation of energy again. The total mechanical energy of the ball at the point 7.97 m below its highest point is equal to the sum of its kinetic energy at that point plus its potential energy at that point.
Let's denote this point as point P:
Total Mechanical Energy at P = KE_P + PE_P
The potential energy at point P is given by:
PE_P = m * g * h_P
Where:
m = mass of the golf ball = 0.0505 kg
g = acceleration due to gravity = 9.8 m/s^2
h_P = height at point P = 23.2 m - 7.97 m = 15.23 m
Substituting the values into the formula:
PE_P = 0.0505 kg * 9.8 m/s^2 * 15.23 m
PE_P ≈ 7.45 J (to two decimal places)
Since the total mechanical energy is conserved, we can say that:
Total Mechanical Energy at P = KE_P + 7.45 J
We already know the total mechanical energy at the highest point is approximately 82.54 J. Therefore:
Total Mechanical Energy at P = KE_highest point + 7.45 J
82.54 J = KE_highest point + 7.45 J
Let's solve for KE_highest point:
KE_highest point = 82.54 J - 7.45 J
KE_highest point ≈ 75.09 J (to two decimal places)
Now, we can determine the speed (v_P) of the ball at point P using the formula for kinetic energy:
KE_P = (1/2) * m * v_P^2
Where:
m = mass of the golf ball = 0.0505 kg
v_P = speed of the golf ball at point P (what we need to find)
Rearranging the formula:
v_P^2 = (2 * KE_P) / m
v_P = sqrt((2 * KE_P) / m)
Substituting the values into the formula:
v_P = sqrt((2 * 75.09 J) / 0.0505 kg)
v_P ≈ 33.16 m/s (to two decimal places)
Therefore, the speed of the ball when it is 7.97 m below its highest point is approximately 33.16 m/s (to two decimal places).
To solve this problem, we need to understand the concepts of kinetic energy and conservation of energy.
(a) To find the kinetic energy at the highest point, we need to consider the conservation of mechanical energy. The total mechanical energy at any point is the sum of kinetic energy (KE) and potential energy (PE).
At the highest point, the golf ball is at its maximum height. At this point, the velocity is zero, and all its initial kinetic energy is now converted into potential energy.
Potential energy (PE) can be calculated using the formula:
PE = m * g * h
Where:
m = mass of the golf ball = 50.5 g (convert to kg by dividing by 1000)
g = acceleration due to gravity = 9.8 m/s^2
h = height = 23.2 m
PE = (50.5/1000) * 9.8 * 23.2
Now, since the kinetic energy (KE) is zero at the highest point, the total mechanical energy is only potential energy:
KE at highest point = 0 J
PE at highest point = (50.5/1000) * 9.8 * 23.2 J
Therefore, the kinetic energy of the ball at its highest point is 0 J.
(b) Now, we need to find the speed of the ball when it is 7.97 m below its highest point. We can use the principle of conservation of energy again. At this point, the potential energy will have reduced, and the kinetic energy will increase.
The initial potential energy (PEi) at the highest point is given by the formula:
PEi = m * g * hi
Where:
m = mass of the golf ball = 50.5 g (convert to kg by dividing by 1000)
g = acceleration due to gravity = 9.8 m/s^2
hi = initial height = 23.2 m
PEi = (50.5/1000) * 9.8 * 23.2
Now, at a height of 7.97 m below the highest point, the potential energy (PEf) will be given by the formula:
PEf = m * g * hf
Where:
m = mass of the golf ball = 50.5 g (convert to kg by dividing by 1000)
g = acceleration due to gravity = 9.8 m/s^2
hf = final height = (23.2 - 7.97) m
PEf = (50.5/1000) * 9.8 * (23.2 - 7.97)
Since the total mechanical energy is conserved, the change in potential energy will be equal to the change in kinetic energy:
ΔKE = PEi - PEf
Therefore, the change in kinetic energy will be equal to the change in potential energy.
Finally, we can calculate the final kinetic energy (KEf) using the formula:
KEf = PEi - ΔKE
Now, since kinetic energy (KE) is given by the formula:
KE = 0.5 * m * v^2
We can solve for the speed (v) by rearranging the formula:
v = sqrt(2 * KE / m)
Substituting the values, we have:
KEf = PEi - (PEi - PEf)
v = sqrt(2 * KEf / m)
Using these equations, you can calculate the final kinetic energy and the speed of the ball when it is 7.97 m below its highest point.